Since
PE=3/6size 12{P left (E right )=3/6} {} , and
PF=2/6size 12{P left (F right )=2/6} {} , a student may say
PE∪F=3/6+2/6size 12{P left (E union F right )=3/6+2/6} {} . This will be incorrect because the element 6, which is in both
Esize 12{E} {} and
Fsize 12{F} {} has been counted twice, once as an element of
Esize 12{E} {} and once as an element of
Fsize 12{F} {} . In other words, the set
E∪Fsize 12{E union F} {} has only four elements and not five. Therefore,
PE∪F=4/6size 12{P left (E union F right )=4/6} {} and not
5/6size 12{5/6} {} .

Finding the probability of
E∪F , is the same as finding the probability that
E will happen, or
F will happen, or both will happen. If we count the number of elements
n(E) in
E , and add to it the number of elements
n(F) in
F , the points in both
E and
F are counted twice, once as elements of
E and once as elements of
F . Now if we subtract from the sum,
n(E)+n(F) , the number
n(E∩F) , we remove the duplicity and get the correct answer. So as a rule,

n(E∪F)=n(E)+n(F)–n(E∩F)

By dividing the entire equation by
n(S) , we get

n(E∪F)n(S)=n(E)n(S)+n(F)n(S)–n(E∩F)n(S)

Since the probability of an event is the number of elements in that event divided by the number of all possible outcomes, we have

P(E∪F)=P(E)+P(F)–P(E∩F)

Applying the above for this example, we get

P(E∪F)=3/6+2/6-1/6=4/6

This is because, when we add
P(E) and
P(F) , we have added
P(E∩F) twice. Therefore, we
must subtract
P(E∩F) , once.

This gives us the general formula, called the
Addition Rule , for finding the probability of the
union of two events. It states

P(E∪F)=P(E)+P(F)–P(E∩F)

If two events E and F are mutually exclusive, then
E∩F=∅ and
P(E∩F)=0 , and we get

If a card is drawn from a deck, use the addition rule to find the probability of obtaining an ace or a heart.

Let
Asize 12{A} {} be the event that the card is an ace, and
Hsize 12{H} {} the event that it is a heart.

Since there are four aces, and thirteen hearts in the deck,
PA=4/52size 12{P left (A right )=4/"52"} {} and
PH=13/52size 12{P left (H right )="13"/"52"} {} . Furthermore, since the intersection of two events is an ace of hearts,
PA∩H=1/52size 12{P left (A intersection H right )=1/"52"} {}

We need to find
PA∪Hsize 12{P left (A union H right )} {} .

PA∪H=PA+PH–PA∩H=4/52+13/52−1/52=16/52size 12{P left (A union H right )=P left (A right )+P left (H right )–P left (A intersection H right )=4/"52"+"13"/"52" - 1/"52"="16"/"52"} {} .

Two dice are rolled, and the events
Fsize 12{F} {} and
Tsize 12{T} {} are as follows:

F=The sum of the dice is foursize 12{F= left lbrace "The sum of the dice is four" right rbrace } {} and
T=At least one die shows a threesize 12{T= left lbrace "At least one die shows a three" right rbrace } {}

F=1,3,2,2,3,1size 12{F= left lbrace left (1,3 right ), left (2,2 right ), left (3,1 right ) right rbrace } {}

T=3,1,3,2,3,3,3,4,3,5,3,6,1,3,2,3,4,3,5,3,6,3size 12{T= left lbrace left (3,1 right ), left (3,2 right ), left (3,3 right ), left (3,4 right ), left (3,5 right ), left (3,6 right ), left (1,3 right ), left (2,3 right ), left (4,3 right ), left (5,3 right ), left (6,3 right ) right rbrace } {}

F∩T=1,3,3,1size 12{F intersection T= left lbrace left (1,3 right ), left (3,1 right ) right rbrace } {}

Since
PF∪T=PF+PT−PF∩Tsize 12{P left (F union T right )=P left (F right )+P left (T right ) - P left (F intersection T right )} {}

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.

8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?

Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?